Need more vertices for weight paint
There is a soduku of issues here and I don't know what to address specifically, so I'm going to skim all of them.

The steps to model and rig a hand are:

1) Model the hand.  You seem to be running into a topology issue here.  When people talk about "Topology" in terms of 3d modeling, they are talking about the layout and organization of the vertices and edges.  The rule of thumb is that each vertex and edge should be describe something unique about the model.  I could probably dissolve about 2/3rds of the verts in that hand and it would look functionally the same.  Additionally, there is isn't really topology describing how the fingers would bend.

The following wouldn't be considered good topology or even a good hand, but it is functional as far as animating is concerned.  Notice that even without a texture, you can tell where the fingers will bend and that it is a monster-y hand. 



It is also only 500 verts and faces, which is far less than your hand right now.    

2) Make the armature.  Rigging is an entire profession, so to keep it short: create an armature and add bones until you are satisfied you have enough control points. You seem to have done that, though I will add you seem to have extra bones in the palm.   

3)Bind the armature to the Mesh.  In Blender you do that with the Armature modifier on the mesh.  Add the modifier to the mesh, then specify the Armature you want to use in the modifier.


You can also click the mesh, than the armature while holding shift, then press ctrl+P and choose the Armature Deform option of your liking, this does the same thing.



4) Tweak bone weights.  Now you get to specify how each bone affects each vertex in the mesh.  You will need to turn off "lock edit modes" in Blender 2.8x, as it prevents being in pose mode while weight painting.  



Now: select your armature, and change to pose mode.  Then, select your mesh, and go into weight paint mode.  From here, you should be able to click on a bone to select it.  Click and drag on the mesh to apply your current tool. You can change your current tool by pressing N to bring up the toolshelf, going to the "tool" tab, and clicking the picture to change what brush you are using.  Adding to the mesh increases the proportion of the influence exerted, and the color will change from dark blue (no influence) -> green -> yellow -> red (full influence).  You can try moving the selected bone by pressing  R to test how it is working out.



There are plenty of online resources that can describe this in detail, but that is the gist.     

Update
It appears that there are so many influences on these vertices that weight paint is giving up.  Four is pushing it, but many of these vertices have at least 7.  

   

Completely blanking these influences allows weight paint to work again, but honestly, deleting and re-making the offending geometry will probably be easier and cleaner.  You do not need this many verts.